On the logarithmic equilibrium measure on curves

Abstract

Let $\mu$ be the logarithmic equilibrium measure on a compact set $\gamma \subset \mathbb{R}^{d}$. We prove that $\mu$ is absolutely continuous with respect to the length measure on the part of $\gamma$ which can be locally expressed as the graph of a $C^{1,\alpha}$-function $\mathbb{R} \to \mathbb{R}^{d - 1}$, $\alpha > 0$. For $d = 2$, at least in the case where $\gamma$ is a compact $C^{1,\alpha}$-graph, our result can also be deduced from the classical fact that $\mu$ coincides with the harmonic measure of $\Omega =\mathbb{R}^{2} \, \setminus \, \gamma$ with pole at $\infty$. For $d \geq 3$, however, our result is new even for $C^{\infty}$-graphs. In fact, up to now it was not even known if the support of $\mu$ has positive dimension.